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464 Chapter 8 IIR FILTER DESIGN
The phase-delay of a system is defined as Φ(ω) △ =
−̸
H ( e jω) /ω and is measured in samples.
It can be shown that if we choose
( ) 2 − d
(2 − d)(1 − d)
a 1 =1 , a 2 =
1+d
(2 + d)(1 + d)
(8.80)
Then phase-delay Φ(ω) atlow frequencies is approximated by d in samples. Verify this
result by plotting Φ(ω) over −π/2 ≤ ω ≤ π/2 for d =0.1, d =0.05, and d =0.01.
P8.12 Design an analog Butterworth lowpass filter that has a 0.25 dB or better ripple at
500 rad/sec and at least 50 dB of attenuation at 2000 rad/sec. Determine the system
function in a rational function form. Plot the magnitude response, the log-magnitude
response in dB, the phase response, and the impulse response of the filter.
P8.13 Design an analog Butterworth lowpass filter that has a 0.5 dBorbetter ripple at 10 kHz
and at least 45 dB of attenuation at 20 kHz. Determine the system function in a cascade
form. Plot the magnitude response, the log-magnitude response in dB, the group-delay, and
the impulse response of the filter.
P8.14 Design a lowpass analog Chebyshev-I filter with an acceptable ripple of 1 dB for |Ω| ≤10
and an attenuation of 50 dB or greater beyond |Ω| =15rad/sec. Determine the system
function in a rational function form. Plot the magnitude response, the log-magnitude
response in dB, the group-delay, and the impulse response of the filter.
P8.15 Design a lowpass analog Chebyshev-I filter with the following characteristics:
• A passband ripple of 0.5 dB,
• passband cutoff frequency of 4 kHz, and
• stopband attenuation of 45 dB or greater beyond 20 kHz.
Determine the system function in a cascade form. Plot the magnitude response, the
log-magnitude response in dB, the phase response, and the impulse response of the filter.
P8.16 A signal x a(t) contains two frequencies, 10 kHz and 15 kHz. We want to suppress the
15 kHz component to 50 dB attenuation while passing the 10 kHz component with less than
0.25 dB attenuation. Design a minimum-order Chebyshev-II analog filter to perform this
filtering operation. Plot the log-magnitude response, and verify the design.
P8.17 Design an analog Chebyshev-II lowpass filter that has a 0.25 dB or better ripple at 250 Hz
and at least 40 dB of attenuation at 400 Hz. Plot the magnitude response, the
log-magnitude response in dB, the group-delay, and the impulse response of the filter.
P8.18 A signal x a(t) contains two frequencies, 10 kHz and 15 kHz. We want to suppress the
15 kHz component to 50 dB attenuation while passing the 10 kHz component with less than
0.25 dB attenuation. Design a minimum-order elliptic analog filter to perform this filtering
operation. Plot the log-magnitude response and verify the design. Compare your design
with the Chebyshev-II design in Problem P8.16.
P8.19 Design an analog elliptic lowpass filter that has a 0.25 dB or better ripple at 500 rad/sec
and at least 50 dB of attenuation at 2000 rad/sec. Determine the system function in a
rational function form. Plot the magnitude response, the log-magnitude response in dB, the
phase response, and the impulse response of the filter. Compare your design with the
Butterworth design in Problem P8.12.
P8.20 Write a MATLAB function to design analog lowpass filters. The format of this function
should be
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